| Résume | While the model theory of henselian valued fields in residue characteristic zero is completely understood, the situation in positive characteristic is rather more subtle, even for local fields like the formal Laurent series field F_p((t)). This applies even when analysing only existential theories, which are arguably of the strongest interest in arithmetic, cf. Hilbert's Tenth Problem. Notable progress was made here in particular by Anscombe–Fehm, who showed that the existential theory of the ring F_p((t)) without parameters is decidable, and Denef–Schoutens, who showed the same when allowing the parameter t assuming Resolution of Singularities. The latter result was later improved in joint work of mine with Anscombe–Fehm. I will report on ongoing work in this direction, focussing on existential theories of henselian valued fields like K((t)) for some base field K of positive characteristic with parameters from a trivially valued base field. As an application, it is possible to find wide classes of function fields F such that it is decidable which polynomial equations over F have solutions in almost all completions of F, as well as stronger results under a Resolution of Singularities assumption. Some of this was first explored in joint work with Fehm.
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